Figure REF  explores the performance of the power-based Eq.(REF ) and logarithm-based Eq.(REF ) in the subcritical, critical, and supercritical flow regimes, and compares their performance against experimental and numerical results. The training data for Eq.(REF ) and Eq.(REF ) was limited to \(Re\)  up to \(2\times 10^5\) . There is a significant discrepancy between the different experimental results, for different reasons, such as the turbulence intensity the positions of the sensors around the sphere [1]}. Eq.(REF ) follows the anticipated trend in the critical flow regime in which the \(C_D\)  is decreasing with increasing \(Re\) . Note that on the contrary, the value of \(C_D\)  from the correlation of Brown and
Lawler [2]} stays constant for \(Re\)  values higher than \(10^4\) . The onset of the critical flow regime for the power-based equation Eq.(REF ) starts at approximately \(Re\approx 10^5\) , earlier than most experimental and numerical results, except the experimental data of Maxworthy [3]}, in which the critical flow regime starts at much lower \( Re\) . At approximately \(Re = 3\times 10^5\)  Eq.(REF ) drops to zero, and its values resembles the experimental values of Achenbach [4]}. The drop of Eq.(REF ) to zero at \(Re\)  = \(3\times 10^5\)  was already predicted algebraically in the previous section, and Eq.(REF ) is the first in literature that predicts with good accuracy the value of \(Re_{cr}\)  reported by the experiments of Achenbach [4]}. From Figure REF , we can see that even the high fidelity simulations of Gerier [6]} with fine grid failed to predict \(Re_{cr}\)  since they failed to resolve the Kolmogorov length scale at such high \(Re\) . The numerical results for the medium grid scheme of Gerier et al. [6]} are close to the predictions of the Eq.(REF ) for \(Re\)  until the critical Reynolds number. The logarithmic based equation (REF ) predicts the onset of the critical flow regime with great accuracy since it follows the \(C_D\)  values from the experiments of Suryanarayana et al.
[8]}, and Achenbach [4]} from \(Re\)  = \(5\times 10^4\)  to about \(3\times 10^5\) . Eq.(REF ) does not does not drop to zero at \(Re_{cr}\)  as Eq.(REF ), however it follows very closely the high fidelity numerical results of Gerier et al. [6]}, for the coarse grid case for \(Re\)  up to \(10^6\) . This shows that Eq.(REF ) follows an approximately physical reality for \(Re\)  up to \(10^6\) , since the results of Gerier et al.
[6]} are generated by solving an approximate form of the Navier-Stokes equations.  Both Eq.(REF ) and Eq.(REF ) fail to predict the increase of \(C_D\)  after the end of the critical flow regime, and the start of the supercritical flow regime at which the boundary layer attached at the surface of the sphere changes from being partly laminar to being fully turbulent. This failure is attributed to the fact the training data used to obtain Eq.(REF ), and Eq.(REF ) are far from the critical flow regime. Predicting \(C_D\)  for the critical flow regime is difficult even for high fidelity solvers. For example, the non-optimized (Nonopt) solver of
Gerier et al. [6]} failed to predict the drag crisis. Instead, it predicts that \(C_D\)  does not change with \(Re\) , similar to what the correlation of Brown and Lawler [2]} predicts. Eq.(REF ), and Eq.(REF ) perform better in the critical regime than the fitting correlation of Morrison [14]} which is a result of fitting experimental data from the literature. Another interesting observation is that the rate of change of \(C_D\)  with \(Re\)  in the critical flow regime, for both Eq.(REF ) and Eq.(REF ), follows the smooth trend similar to the experiments of Maxworthy [3]} and the high fidelity simulations of
Geier et al. [6]}, rather than the sharp nearly discontinuous change of \(C_D\)  observed in the experiments of [4]}, [8]}, [19]}.
